Introduction

Though most investors may not be explicit in saying it, one of their primary goals is increasing the “terminal value” of their investments – i.e., maximizing the growth of an investment over time – for a desired level of risk. Pensions focus on long-term assets relative to liabilities, endowments and foundations on compound returns relative to spending rates, individuals on saving enough for future retirement needs. Yet, professional investors manage portfolios with a goal of maximizing per-period averages per unit of average risk, like Sharpe or information ratios. This raises the question: Is the goal of maximizing measures like Sharpe ratio or information ratio consistent with the goal of maximizing terminal value? The answer is a resounding no.

Managing portfolios to enhance averages – like Sharpe and information ratios – is a very different goal compared to that of enhancing terminal value. The optimality of maximizing Sharpe or information ratios is rooted under the assumption of repeatable single-period investing, while enhancing terminal value is rooted in multi-period investing, where the environment from one period to the next can be very different. The risk of a portfolio over a single period is managed by diversifying holdings – cross-sectional diversification. But once time enters the picture through an emphasis on terminal value, diversifying across time becomes even more important than diversifying across assets. This time diversification is crucial because the level of asset class risk fluctuates, sometimes violently, across time, leading to a form of convexity costs. Most know that convexity costs arise from noise (volatility) around realized returns and result in a drag on compound returns. But what is less known is the fluctuation in risk levels over time, which we call excess risk, amplifies these costs.

While there is little one can do to remove the noise around realized returns to reduce convexity costs, excess portfolio risk can be addressed with time diversification. Diversifying across time reduces the drag on compound returns by decreasing the fluctuation in risk levels to maintain a constant risk level from period to period – a target risk level. The “convexity cost” of this excess risk can be large – resulting in a meaningful reduction in compound return. This is true even if asset returns follow a normal distribution. Portfolio management should seek to reduce this cost by reducing the fluctuations in risk around target levels.

In practice, convexity cost can be greater than many expect as asset return distributions are fat-tailed and the large negative outliers lead to a much larger drag on compound returns than volatility moves. Therefore, portfolio management should develop measures of risk consistent with this reality: tail risk is the most important risk affecting compound returns. The impact on terminal value of just a few outliers, both positive and negative, is much greater than that of many small moves.

We propose an investment approach – which we call adaptive asset allocation – that refocuses the goal of portfolio management toward maximizing terminal value. In this approach, portfolios adapt to current market conditions with an eye toward both the present (cross-sectional diversification) and the future (time diversification).

Interestingly enough, static averages and conventional measures of portfolio efficiency can mask the opportunity presented by adaptive asset allocation: both the information ratio and Sharpe ratio1 may show little advantage to the approach, at least at times. Yet, the positive impact on terminal value can be substantial.

Beginning with the assumption that the goal is to maximize terminal value for a given target risk level, this paper outlines two key concepts that drive terminal value: time diversification and convexity costs, specifically the amplification of these costs due to fat tails. We conclude with a high-level overview of adaptive asset allocation, an investment approach that seeks to enhance terminal value by dynamically managing portfolio outliers.

Time Diversification: A New Dimension in Risk Management

Managing portfolios to maximize terminal value adds a new dimension of risk: time. Terminal value is a reflection of portfolio value growth over multiple periods as returns compound. As a result, risk must be managed across more than just a single period. The impact of risk over time is ignored in most investment models, which focus on a single period. In a single-period model, the only risk that can be managed is cross-sectional risk through holding a diversified portfolio of assets. However, once time enters the picture, risk can also be diversified across time – and the impact of diversifying across time can be much greater than diversifying in the cross section.

To illustrate, consider an investor with $100 million of capital. For 99 days, the investor risks $1 million in a portfolio diversified across multiple assets. Then, on the 100th day, the investor risks all $100 million in the same portfolio. This portfolio is “diversified” in the sense that it has balanced risk across a number of assets in each single period. However, it has failed to balance risk across time. The potential for a 100% loss on the final day results in far greater risk to terminal value than all of the previous days combined.

This idea is highlighted with a more realistic example in Exhibit 1, which shows the risk of hypothetical assets at one point in time on the left and the risk of a portfolio across time on the right. Just as a cross-sectionally diversified portfolio would reduce the relative weight to the riskier Asset 3, so too would a time-diversified portfolio reduce the relative weight to the riskier Time Period 3. Both forms of diversification will reduce the risk around the average, the former across one period and the latter across multiple periods.

Ignoring changes in risk over time can have serious consequences for terminal portfolio values. Even so-called “balanced” portfolios can have dramatically varying risk over time. As an example, rolling standard deviations of a 60/40 portfolio are shown in Exhibit 2. At times, the risk of the strategy is high. At other times, it is low. The highly variable risk of the “balanced” portfolio over time results in a lack of time diversification.2

Current portfolio construction approaches determine allocations without taking account of varying expected risks over time. The result is often wildly changing levels of portfolio risk, which can create a significant drag on performance – and one that is not captured in per-period means and variances. Consider the following simple examples:

Average performance is realized over identical and repeatable periods: Imagine a coin showing heads half the time and tails half the time. With just four flips, it is not so likely that half the flips will be heads. But with 1,000 flips, it is very likely that close to half will be heads. That is the power of repeating the game and repeating the same scenario: you realize the average outcome.

Averages ignore the effects of compounding across non-identical periods: Investors do not have the luxury of living in a single-period, repeatable world – or a world of averages. An investment that gains 50% today and loses 50% tomorrow has an average return equal to zero. But this is not the case in a multi-period compounding world where capital put at risk varies. Gaining 50% today and losing 50% tomorrow yields an overall loss in portfolio value of 25%, very different from the “average return” of zero.

We invest not over repeatable single periods, but across non-identical multiple periods. And investing across multiple periods adds the time dimension to the risk. If one cares about terminal value, one needs to refocus attention and effort away from modeling the averages and toward reducing the variability of portfolio risk from target risk over time. Doing so can reduce portfolio return drag from excess risk, or convexity cost.

Reducing Convexity Costs Can Help Maximize Terminal Value

The time-varying level of risk in a portfolio can significantly reduce compound returns. And most investors make inaccurate assumptions about how changing risk levels can impact terminal value, if at all. Imagine two portfolios with the same expected return targeting a beta of 0.5 to the equity market:

Portfolio A invests fully in U.S. equities 50% of the time and fully holds cash otherwise.

Portfolio B holds 50% of U.S. equities and 50% of cash at all times.

These two portfolios both have an expected return equal to 50% of the expected market return, but will not yield the same terminal value. The risk of Portfolio A changes over time from zero (when only cash is held) to that of the risk of U.S. equities (when only U.S. equities are held). While both have a 0.5 beta, the fluctuation in risk causes the volatility of Portfolio A across time to be 70% of U.S. equities and not one-half. And it is this excess volatility that results in convexity cost. This cost can be avoided by time diversification. Portfolio B is a time-diversified strategy whose risk stays at 50% of the risk of U.S. equities at all times and hence doesn’t suffer the convexity drag.

As shown in Exhibit 3, the drag is significant: Portfolio B compounds wealth 50 basis points (bps) per year faster than Portfolio A. Over 87 years, $1,000 invested in Portfolio A grows to $49,000, while $1,000 in Portfolio B grows to $70,000. The power of time-series diversification is that terminal value is enhanced without requiring the ability to predict expected returns. All that is required is to reduce the variability of risk around a target level. And in this case, that translates to maintaining a fixed exposure to the U.S. market versus varying exposure over time.